A331567 - OEIS

A331567

Array read by antidiagonals: A(n,k) is the number of binary matrices with k columns and any number of distinct nonzero rows with n ones in every column.

%I #16 Jan 25 2020 17:55:07

%S 1,1,1,1,1,1,1,3,0,1,1,13,6,0,1,1,75,120,0,0,1,1,541,6174,1104,0,0,1,

%T 1,4683,449520,413088,5040,0,0,1,1,47293,49686726,329520720,18481080,

%U 0,0,0,1,1,545835,7455901320,491236986720,179438982360,522481680,0,0,0,1

%N Array read by antidiagonals: A(n,k) is the number of binary matrices with k columns and any number of distinct nonzero rows with n ones in every column.

%H Andrew Howroyd, <a href="/A331567/b331567.txt">Table of n, a(n) for n = 0..209</a>

%F A(n,k) = 0 for k > 0, n > 2^(k-1).

%F A(2^(k-1), k) = (2^k-1)! for k > 0.

%F A331643(n) = Sum_{d|n} A(n/d, d).

%e Array begins:

%e ===============================================================

%e n\k | 0 1 2 3 4 5 6

%e ----+----------------------------------------------------------

%e 0 | 1 1 1 1 1 1 1 ...

%e 1 | 1 1 3 13 75 541 4683 ...

%e 2 | 1 0 6 120 6174 449520 49686726 ...

%e 3 | 1 0 0 1104 413088 329520720 491236986720 ...

%e 4 | 1 0 0 5040 18481080 179438982360 3785623968170400 ...

%e 5 | 1 0 0 0 522481680 70302503250720 ...

%e 6 | 1 0 0 0 7875584640 ...

%e ...

%e The A(2,2) = 6 matrices are:

%e [1 1] [1 1] [1 0] [1 0] [0 1] [0 1]

%e [1 0] [0 1] [1 1] [0 1] [1 1] [1 0]

%e [0 1] [1 0] [0 1] [1 1] [1 0] [1 1]

%o (PARI)

%o WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}

%o D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); WeighT(v)[n]^k/prod(i=1, #v, i^v[i]*v[i]!)}

%o T(n, k)={ my(m=n*k+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))), f=Vec(serlaplace(1/(1+x) + O(x*x^m))/(x-1))); if(n==0, 1, sum(j=1, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*sum(i=j, m, q[i-j+1]*f[i]))); }

%Y Rows n=1..3 are A000670, A331640, A331641.

%Y Column k=5 is A331642.

%Y Cf. A188445, A330942, A331568, A331569, A331571, A331643.

%K nonn,tabl

%O 0,8

%A _Andrew Howroyd_, Jan 20 2020